Statistical Modelling of COVID-19 Outbreak in Italy

07 Apr 2020



Nonlinear growth models

Nonlinear growth models represent an instance of nonlinear regression models, a class of models taking the general form \[ y = \mu(x, \theta) + \epsilon, \] where \(\mu(x, \theta)\) is the mean function which depends on a possibly vector-valued parameter \(\theta\), and a possibly vector-valued predictor \(x\). The stochastic component \(\epsilon\) represents the error with mean zero and constant variance. Usually, a Gaussian distribution is also assumed for the error term.

By defining the mean function \(\mu(x, \theta)\) we may obtain several different models, all characterized by the fact that parameters \(\theta\) enter in a nonlinear way into the equation. Parameters are usually estimated by nonlinear least squares which aims at minimizing the residual sum of squares.

Exponential

\[ \mu(x) = \theta_1 \exp\{\theta_2 x\} \] where \(\theta_1\) is the value at the origin (i.e. \(\mu(x=0)\)), and \(\theta_2\) represents the (constant) relative ratio of change (i.e. \(\frac{d\mu(x)}{dx }\frac{1}{\mu(x)} = \theta_2\)). Thus, the model describes an increasing (exponential growth if \(\theta_2 > 0\)) or decreasing (exponential decay if \(\theta_2 < 0\)) trend with constant relative rate.

Logistic

\[ \mu(x) = \frac{\theta_1}{1+\exp\{(\theta_2 - x)/\theta_3\}} \] where \(\theta_1\) is the upper horizontal asymptote, \(\theta_2\) represents the x-value at the inflection point of the symmetric growth curve, and \(\theta_3\) represents a scale parameter (and \(1/\theta_3\) is the growth-rate parameter that controls how quickly the curve approaches the upper asymptote).

Gompertz

\[ \mu(x) = \theta_1 \exp\{-\theta_2 \theta_3^x\} \] where \(\theta_1\) is the horizontal asymptote, \(\theta_2\) represents the value of the function at \(x = 0\) (displacement along the x-axis), and \(\theta_3\) represents a scale parameter.

The difference between the logistic and Gompertz functions is that the latter is not symmetric around the inflection point.

Richards

\[ \mu(x) = \theta_1 (1 - \exp\{-\theta_2 x\})^{\theta_3} \] where \(\theta_1\) is the horizontal asymptote, \(\theta_2\) represents the rate of growth, and \(\theta_3\) in part determines the point of inflection on the y-axis.

Data

Dipartimento della Protezione Civile: COVID-19 Italia - Monitoraggio della situazione http://arcg.is/C1unv

Source: https://github.com/pcm-dpc/COVID-19

url = "https://raw.githubusercontent.com/pcm-dpc/COVID-19/master/dati-andamento-nazionale/dpc-covid19-ita-andamento-nazionale.csv"
COVID19 <- read.csv(file = url, stringsAsFactors = FALSE)
COVID19$data <- as.Date(COVID19$data)
# DT::datatable(COVID19)

Warnings

- 29/03/2020: dati Regione Emilia Romagna parziali (dato tampone non aggiornato).
- 26/03/2020: dati Regione Piemonte parziali (-50 deceduti - comunicazione tardiva)
- 18/03/2020: dati Regione Campania non pervenuti.
- 18/03/2020: dati Provincia di Parma non pervenuti.
- 17/03/2020: dati Provincia di Rimini non aggiornati
- 16/03/2020: dati P.A. Trento e Puglia non pervenuti.
- 11/03/2020: dati Regione Abruzzo non pervenuti.
- 10/03/2020: dati Regione Lombardia parziali.
- 07/03/2020: dati Brescia +300 esiti positivi


Modelling total infected

# create data for analysis
data = data.frame(date = COVID19$data,
                  y = COVID19$totale_casi)
data$x = as.numeric(data$date) - min(as.numeric(data$date)) + 1
DT::datatable(data, options = list("pageLength" = 5))

Estimation

Exponential

mod1_start = lm(log(y) ~ x, data = data)
b = unname(coef(mod1_start))
start = list(th1 = exp(b[1]), th2 = b[2])
exponential <- function(x, th1, th2) th1 * exp(th2 * x)
mod1 = nls(y ~ exponential(x, th1, th2), data = data, start = start)
summary(mod1)
## 
## Formula: y ~ exponential(x, th1, th2)
## 
## Parameters:
##        Estimate  Std. Error t value      Pr(>|t|)    
## th1 6758.718857  913.938740   7.395 0.00000000398 ***
## th2    0.071650    0.003541  20.233       < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 10280 on 42 degrees of freedom
## 
## Number of iterations to convergence: 10 
## Achieved convergence tolerance: 0.000009694

Logistic

mod2 = nls(y ~ SSlogis(x, Asym, xmid, scal), data = data)
summary(mod2)
## 
## Formula: y ~ SSlogis(x, Asym, xmid, scal)
## 
## Parameters:
##         Estimate  Std. Error t value Pr(>|t|)    
## Asym 147850.4983   1618.4196   91.36   <2e-16 ***
## xmid     30.8837      0.1823  169.42   <2e-16 ***
## scal      6.0388      0.1008   59.94   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1408 on 41 degrees of freedom
## 
## Number of iterations to convergence: 0 
## Achieved convergence tolerance: 0.0000002824

Gompertz

mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data)
# start = list(Asym = coef(mod2)[1])
# tmp = list(y = log(log(start$Asym) - log(data$y)), x = data$x)
# b = unname(coef(lm(y ~ x, data = tmp)))
# start = c(start, c(b2 = exp(b[1]), b3 = exp(b[2])))
# mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data, start = start,
#            control = nls.control(maxiter = 1000))
summary(mod3)
## 
## Formula: y ~ SSgompertz(x, Asym, b2, b3)
## 
## Parameters:
##           Estimate    Std. Error t value Pr(>|t|)    
## Asym 196687.320590   3441.169949   57.16   <2e-16 ***
## b2       10.025135      0.271430   36.93   <2e-16 ***
## b3        0.927422      0.001382  670.94   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1020 on 41 degrees of freedom
## 
## Number of iterations to convergence: 0 
## Achieved convergence tolerance: 0.0000007416

Richards

richards <- function(x, th1, th2, th3) th1*(1 - exp(-th2*x))^th3
Loss  <- function(th, y, x) sum((y - richards(x, th[1], th[2], th[3]))^2) 
start <- optim(par = c(coef(mod2)[1], 0.001, 1), fn = Loss, 
               y = data$y, x = data$x)$par
names(start) <- c("th1", "th2", "th3")
mod4 = nls(y ~ richards(x, th1, th2, th3), data = data, start = start,
           # trace = TRUE, algorithm = "plinear", 
           control = nls.control(maxiter = 1000, tol = 0.1))
# algorithm is not converging... 
summary(mod4)
## 
## Formula: y ~ richards(x, th1, th2, th3)
## 
## Parameters:
##          Estimate    Std. Error t value Pr(>|t|)    
## th1 214735.824697   7005.475001   30.65   <2e-16 ***
## th2      0.062719      0.002531   24.78   <2e-16 ***
## th3      6.865356      0.359753   19.08   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1330 on 41 degrees of freedom
## 
## Number of iterations to convergence: 3 
## Achieved convergence tolerance: 0.08979
# library(nlmrt)
# mod4 = nlxb(y ~ th1*(1 - exp(-th2*x))^th3, 
#             data = data, start = start, trace = TRUE)

Models comparison

models = list("Exponential model" = mod1, 
              "Logistic model" = mod2, 
              "Gompertz model" = mod3,
              "Richards model" = mod4)
tab = data.frame(loglik = sapply(models, logLik),
                 df = sapply(models, function(m) attr(logLik(m), "df")),
                 Rsquare = sapply(models, function(m) 
                                  cor(data$y, fitted(m))^2),
                 AIC = sapply(models, AIC),
                 AICc = sapply(models, AICc),
                 BIC = sapply(models, BIC))
sel <- apply(tab[,4:6], 2, which.min)
tab$"" <- sapply(tabulate(sel, nbins = length(models))+1, symnum,
                 cutpoints = 0:4, symbols = c("", "*", "**", "***"))
knitr::kable(tab)
loglik df Rsquare AIC AICc BIC
Exponential model -467.8758 3 0.9591878 941.7516 942.3516 947.1042
Logistic model -379.8783 4 0.9992512 767.7565 768.7822 774.8933
Gompertz model -365.6777 4 0.9996052 739.3554 740.3811 746.4922 ***
Richards model -377.3526 4 0.9993631 762.7053 763.7309 769.8420 
ggplot(data, aes(x = date, y = y)) + 
  geom_point() +
  geom_line(aes(y = fitted(mod1), color = "Exponential")) +
  geom_line(aes(y = fitted(mod2), color = "Logistic")) +
  geom_line(aes(y = fitted(mod3), color = "Gompertz")) +
  geom_line(aes(y = fitted(mod4), color = "Richards")) +
  labs(x = "", y = "Infected", color = "Model") +
  scale_color_manual(values = cols) +
  scale_y_continuous(breaks = seq(0, coef(mod2)[1], by = 10000),
                     minor_breaks = seq(0, coef(mod2)[1], by = 5000)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

last_plot() +
  scale_y_continuous(trans = "log10", limits = c(100,NA)) +
  labs(y = "Infected (log10 scale)")

Predictions

Point estimates

df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1),
               fit1 = predict(mod1, newdata = df),
               fit2 = predict(mod2, newdata = df),
               fit3 = predict(mod3, newdata = df),
               fit4 = predict(mod4, newdata = df))
ylim = c(0, max(df[,c("fit2", "fit3")]))
ggplot(data, aes(x = date, y = y)) + 
  geom_point() +
  geom_line(data = df, aes(x = date, y = fit1, color = "Exponential")) +
  geom_line(data = df, aes(x = date, y = fit2, color = "Logistic")) +
  geom_line(data = df, aes(x = date, y = fit3, color = "Gompertz")) +
  geom_line(data = df, aes(x = date, y = fit4, color = "Richards")) +
  coord_cartesian(ylim = ylim) +
  labs(x = "", y = "Infected", color = "Model") +
  scale_y_continuous(breaks = seq(0, max(ylim), by = 10000),
                     minor_breaks = seq(0, max(ylim), by = 5000)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  scale_color_manual(values = cols) +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

Prediction intervals

# compute prediction using Moving Block Bootstrap (MBB) for nls
df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1))

pred1 = cbind(df, "fit" = predict(mod1, newdata = df))
pred1[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod1, df[df$x > max(data$x),])[,2:3]

pred2 = cbind(df, "fit" = predict(mod2, newdata = df))
pred2[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod2, df[df$x > max(data$x),])[,2:3]

pred3 = cbind(df, "fit" = predict(mod3, newdata = df))
pred3[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod3, df[df$x > max(data$x),])[,2:3]

pred4 = cbind(df, "fit" = predict(mod4, newdata = df))
pred4[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod4, df[df$x > max(data$x),])[,2:3]

# predictions for next day
pred = rbind(subset(pred1, x == max(data$x)+1, select = 2:5),
             subset(pred2, x == max(data$x)+1, select = 2:5),
             subset(pred3, x == max(data$x)+1, select = 2:5),
             subset(pred4, x == max(data$x)+1, select = 2:5))
print(pred, digits = 3)
##           date    fit    lwr    upr
## 45  2020-04-08 169878 140024 200333
## 451 2020-04-08 134831 131209 137708
## 452 2020-04-08 140313 137623 143208
## 453 2020-04-08 140965 137519 145191

ylim = c(0, max(pred2$upr, pred3$upr, na.rm=TRUE))
ggplot(data, aes(x = date, y = y)) + 
  geom_point() +
  geom_line(data = pred1, aes(x = date, y = fit, color = "Exponential")) +
  geom_line(data = pred2, aes(x = date, y = fit, color = "Logistic")) +
  geom_line(data = pred3, aes(x = date, y = fit, color = "Gompertz")) +
  geom_line(data = pred4, aes(x = date, y = fit, color = "Richards")) +
  geom_ribbon(data = pred1, aes(x = date, ymin = lwr, ymax = upr), 
              inherit.aes = FALSE, fill = cols[1], alpha=0.3) +
  geom_ribbon(data = pred2, aes(x = date, ymin = lwr, ymax = upr), 
              inherit.aes = FALSE, fill = cols[2], alpha=0.3) +
  geom_ribbon(data = pred3, aes(x = date, ymin = lwr, ymax = upr),
              inherit.aes = FALSE, fill = cols[3], alpha=0.3) +
  geom_ribbon(data = pred4, aes(x = date, ymin = lwr, ymax = upr),
              inherit.aes = FALSE, fill = cols[4], alpha=0.3) +
  coord_cartesian(ylim = c(0, max(ylim))) +
  labs(x = "", y = "Infected", color = "Model") +
  scale_y_continuous(minor_breaks = seq(0, max(ylim), by = 10000)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  scale_color_manual(values = cols) +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

Modelling total deceased

# create data for analysis
data = data.frame(date = COVID19$data,
                  y = COVID19$deceduti)
data$x = as.numeric(data$date) - min(as.numeric(data$date)) + 1
DT::datatable(data, options = list("pageLength" = 5))

Estimation

Exponential

mod1_start = lm(log(y) ~ x, data = data)
b = unname(coef(mod1_start))
start = list(th1 = exp(b[1]), th2 = b[2])
exponential <- function(x, th1, th2) th1 * exp(th2 * x)
mod1 = nls(y ~ exponential(x, th1, th2), data = data, start = start)
summary(mod1)
## 
## Formula: y ~ exponential(x, th1, th2)
## 
## Parameters:
##       Estimate Std. Error t value     Pr(>|t|)    
## th1 472.714054  69.681862   6.784 0.0000000296 ***
## th2   0.084798   0.003775  22.462      < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1092 on 42 degrees of freedom
## 
## Number of iterations to convergence: 11 
## Achieved convergence tolerance: 0.000002829

Logistic

mod2 = nls(y ~ SSlogis(x, Asym, xmid, scal), data = data)
summary(mod2)
## 
## Formula: y ~ SSlogis(x, Asym, xmid, scal)
## 
## Parameters:
##         Estimate  Std. Error t value Pr(>|t|)    
## Asym 19579.34504   228.11191   85.83   <2e-16 ***
## xmid    33.74339     0.17071  197.67   <2e-16 ***
## scal     5.62057     0.08452   66.50   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 146.1 on 41 degrees of freedom
## 
## Number of iterations to convergence: 0 
## Achieved convergence tolerance: 0.0000005029

Gompertz

mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data)
# manually set starting values
# start = list(Asym = coef(mod2)[1])
# tmp = list(y = log(log(start$Asym) - log(data$y)), x = data$x)
# b = unname(coef(lm(y ~ x, data = tmp)))
# start = c(start, c(b2 = exp(b[1]), b3 = exp(b[2])))
# mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data, start = start, 
#            control = nls.control(maxiter = 10000))
summary(mod3)
## 
## Formula: y ~ SSgompertz(x, Asym, b2, b3)
## 
## Parameters:
##          Estimate   Std. Error t value Pr(>|t|)    
## Asym 28582.190126   571.205701   50.04   <2e-16 ***
## b2      13.401511     0.376627   35.58   <2e-16 ***
## b3       0.928034     0.001307  710.00   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 103.3 on 41 degrees of freedom
## 
## Number of iterations to convergence: 0 
## Achieved convergence tolerance: 0.0000002748

Richards

richards <- function(x, th1, th2, th3) th1*(1 - exp(-th2*x))^th3
Loss  <- function(th, y, x) sum((y - richards(x, th[1], th[2], th[3]))^2) 
start <- optim(par = c(coef(mod2)[1], 0.001, 1), fn = Loss, 
               y = data$y, x = data$x)$par
names(start) <- c("th1", "th2", "th3")
mod4 = nls(y ~ richards(x, th1, th2, th3), data = data, start = start,
           # trace = TRUE, algorithm = "port", 
           control = nls.control(maxiter = 1000))
summary(mod4)
## 
## Formula: y ~ richards(x, th1, th2, th3)
## 
## Parameters:
##         Estimate   Std. Error t value Pr(>|t|)    
## th1 30999.994563  1024.197584   30.27   <2e-16 ***
## th2     0.064717     0.002202   29.38   <2e-16 ***
## th3     9.709261     0.477609   20.33   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 127.6 on 41 degrees of freedom
## 
## Number of iterations to convergence: 2 
## Achieved convergence tolerance: 0.000001331

Models comparison

models = list("Exponential model" = mod1, 
              "Logistic model" = mod2, 
              "Gompertz model" = mod3,
              "Richards model" = mod4)
tab = data.frame(loglik = sapply(models, logLik),
                 df = sapply(models, function(m) attr(logLik(m), "df")),
                 Rsquare = sapply(models, function(m) 
                                  cor(data$y, fitted(m))^2),
                 AIC = sapply(models, AIC),
                 AICc = sapply(models, AICc),
                 BIC = sapply(models, BIC))
sel <- apply(tab[,4:6], 2, which.min)
tab$"" <- sapply(tabulate(sel, nbins = length(models))+1, symnum,
                 cutpoints = 0:4, symbols = c("", "*", "**", "***"))
knitr::kable(tab)
loglik df Rsquare AIC AICc BIC
Exponential model -369.2251 3 0.9704563 744.4502 745.0502 749.8027
Logistic model -280.1774 4 0.9995129 568.3548 569.3805 575.4916
Gompertz model -264.9186 4 0.9997137 537.8372 538.8628 544.9739 ***
Richards model -274.2330 4 0.9995791 556.4661 557.4917 563.6028 
ggplot(data, aes(x = date, y = y)) + 
  geom_point() +
  geom_line(aes(y = fitted(mod1), color = "Exponential")) +
  geom_line(aes(y = fitted(mod2), color = "Logistic")) +
  geom_line(aes(y = fitted(mod3), color = "Gompertz")) +
  geom_line(aes(y = fitted(mod4), color = "Richards")) +
  labs(x = "", y = "Deceased", color = "Model") +
  scale_color_manual(values = cols) +
  scale_y_continuous(breaks = seq(0, coef(mod2)[1], by = 1000),
                     minor_breaks = seq(0, coef(mod2)[1], by = 500)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

last_plot() +
  scale_y_continuous(trans = "log10", limits = c(10,NA)) +
  labs(y = "Deceased (log10 scale)")

Predictions

Point estimates

df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1),
               fit1 = predict(mod1, newdata = df),
               fit2 = predict(mod2, newdata = df),
               fit3 = predict(mod3, newdata = df),
               fit4 = predict(mod4, newdata = df))
ylim = c(0, max(df[,-(1:3)]))
ggplot(data, aes(x = date, y = y)) + 
  geom_point() +
  geom_line(data = df, aes(x = date, y = fit1, color = "Exponential")) +
  geom_line(data = df, aes(x = date, y = fit2, color = "Logistic")) +
  geom_line(data = df, aes(x = date, y = fit3, color = "Gompertz")) +
  geom_line(data = df, aes(x = date, y = fit4, color = "Richards")) +
  coord_cartesian(ylim = ylim) +
  labs(x = "", y = "Deceased", color = "Model") +
  scale_y_continuous(breaks = seq(0, max(ylim), by = 1000),
                     minor_breaks = seq(0, max(ylim), by = 1000)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  scale_color_manual(values = cols) +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

Prediction intervals

# compute prediction using Moving Block Bootstrap (MBB) for nls
df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1))

pred1 = cbind(df, "fit" = predict(mod1, newdata = df))
pred1[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod1, df[df$x > max(data$x),])[,2:3]

pred2 = cbind(df, "fit" = predict(mod2, newdata = df))
pred2[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod2, df[df$x > max(data$x),])[,2:3]

pred3 = cbind(df, "fit" = predict(mod3, newdata = df))
pred3[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod3, df[df$x > max(data$x),])[,2:3]

pred4 = cbind(df, "fit" = predict(mod4, newdata = df))
pred4[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod4, df[df$x > max(data$x),])[,2:3]

# predictions for next day
pred = rbind(subset(pred1, x == max(data$x)+1, select = 2:5),
             subset(pred2, x == max(data$x)+1, select = 2:5),
             subset(pred3, x == max(data$x)+1, select = 2:5),
             subset(pred4, x == max(data$x)+1, select = 2:5))
print(pred, digits = 3)
##           date   fit   lwr   upr
## 45  2020-04-08 21470 18322 24945
## 451 2020-04-08 17251 16819 17576
## 452 2020-04-08 17952 17705 18257
## 453 2020-04-08 18018 17705 18424

ylim = c(0, max(pred2$upr, pred3$upr, na.rm=TRUE))
ggplot(data, aes(x = date, y = y)) + 
  geom_point() +
  geom_line(data = pred1, aes(x = date, y = fit, color = "Exponential")) +
  geom_line(data = pred2, aes(x = date, y = fit, color = "Logistic")) +
  geom_line(data = pred3, aes(x = date, y = fit, color = "Gompertz")) +
  geom_line(data = pred4, aes(x = date, y = fit, color = "Richards")) +
  geom_ribbon(data = pred1, aes(x = date, ymin = lwr, ymax = upr), 
              inherit.aes = FALSE, fill = cols[1], alpha=0.3) +
  geom_ribbon(data = pred2, aes(x = date, ymin = lwr, ymax = upr), 
              inherit.aes = FALSE, fill = cols[2], alpha=0.3) +
  geom_ribbon(data = pred3, aes(x = date, ymin = lwr, ymax = upr),
              inherit.aes = FALSE, fill = cols[3], alpha=0.3) +
  geom_ribbon(data = pred4, aes(x = date, ymin = lwr, ymax = upr),
              inherit.aes = FALSE, fill = cols[4], alpha=0.3) +
  coord_cartesian(ylim = c(0, max(ylim))) +
  labs(x = "", y = "Deceased", color = "Model") +
  scale_y_continuous(minor_breaks = seq(0, max(ylim), by = 1000)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  scale_color_manual(values = cols) +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

Modelling recovered

# create data for analysis
data = data.frame(date = COVID19$data,
                  y = COVID19$dimessi_guariti)
data$x = as.numeric(data$date) - min(as.numeric(data$date)) + 1
DT::datatable(data, options = list("pageLength" = 5))

Estimation

Exponential

mod1_start = lm(log(y) ~ x, data = data)
b = unname(coef(mod1_start))
start = list(th1 = exp(b[1]), th2 = b[2])
exponential <- function(x, th1, th2) th1 * exp(th2 * x)
mod1 = nls(y ~ exponential(x, th1, th2), data = data, start = start)
summary(mod1)
## 
## Formula: y ~ exponential(x, th1, th2)
## 
## Parameters:
##       Estimate Std. Error t value       Pr(>|t|)    
## th1 516.642923  60.844938   8.491 0.000000000117 ***
## th2   0.089825   0.002995  29.990        < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1091 on 42 degrees of freedom
## 
## Number of iterations to convergence: 11 
## Achieved convergence tolerance: 0.00000268

Logistic

mod2 = nls(y ~ SSlogis(x, Asym, xmid, scal), data = data)
summary(mod2)
## 
## Formula: y ~ SSlogis(x, Asym, xmid, scal)
## 
## Parameters:
##        Estimate Std. Error t value Pr(>|t|)    
## Asym 32016.1468   839.6911   38.13   <2e-16 ***
## xmid    37.0064     0.3864   95.77   <2e-16 ***
## scal     6.4455     0.1424   45.27   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 277.4 on 41 degrees of freedom
## 
## Number of iterations to convergence: 0 
## Achieved convergence tolerance: 0.000006546

Gompertz

mod3 = nls(y ~ SSgompertz(x, Asym, b2, b3), data = data)
summary(mod3)
## 
## Formula: y ~ SSgompertz(x, Asym, b2, b3)
## 
## Parameters:
##          Estimate   Std. Error t value Pr(>|t|)    
## Asym 64713.687529  3204.047565   20.20   <2e-16 ***
## b2      10.054874     0.250256   40.18   <2e-16 ***
## b3       0.948343     0.001535  617.66   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 176.2 on 41 degrees of freedom
## 
## Number of iterations to convergence: 0 
## Achieved convergence tolerance: 0.000001595

Richards

richards <- function(x, th1, th2, th3) th1*(1 - exp(-th2*x))^th3
Loss  <- function(th, y, x) sum((y - richards(x, th[1], th[2], th[3]))^2) 
start <- optim(par = c(coef(mod2)[1], 0.001, 1), fn = Loss, 
               y = data$y, x = data$x)$par
names(start) <- c("th1", "th2", "th3")
mod4 = nls(y ~ richards(x, th1, th2, th3), data = data, start = start,
           # trace = TRUE, # algorithm = "port", 
           control = nls.control(maxiter = 1000))
summary(mod4)
## 
## Formula: y ~ richards(x, th1, th2, th3)
## 
## Parameters:
##         Estimate   Std. Error t value        Pr(>|t|)    
## th1 94282.651221 10289.484765   9.163 0.0000000000179 ***
## th2     0.035483     0.002668  13.298         < 2e-16 ***
## th3     5.719437     0.312529  18.300         < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 194.4 on 41 degrees of freedom
## 
## Number of iterations to convergence: 11 
## Achieved convergence tolerance: 0.0000009866

Models comparison

models = list("Exponential model" = mod1, 
              "Logistic model" = mod2, 
              "Gompertz model" = mod3,
              "Richards model" = mod4)
tab = data.frame(loglik = sapply(models, logLik),
                 df = sapply(models, function(m) attr(logLik(m), "df")),
                 Rsquare = sapply(models, function(m) 
                                  cor(data$y, fitted(m))^2),
                 AIC = sapply(models, AIC),
                 AICc = sapply(models, AICc),
                 BIC = sapply(models, BIC))
sel <- apply(tab[,4:6], 2, which.min)
tab$"" <- sapply(tabulate(sel, nbins = length(models))+1, symnum,
                 cutpoints = 0:4, symbols = c("", "*", "**", "***"))
knitr::kable(tab)
loglik df Rsquare AIC AICc BIC
Exponential model -369.1897 3 0.9839216 744.3794 744.9794 749.7319
Logistic model -308.3937 4 0.9988963 624.7875 625.8131 631.9243
Gompertz model -288.4324 4 0.9994975 584.8648 585.8904 592.0015 ***
Richards model -292.7633 4 0.9994170 593.5266 594.5523 600.6634 
ggplot(data, aes(x = date, y = y)) + 
  geom_point() +
  geom_line(aes(y = fitted(mod1), color = "Exponential")) +
  geom_line(aes(y = fitted(mod2), color = "Logistic")) +
  geom_line(aes(y = fitted(mod3), color = "Gompertz")) +
  geom_line(aes(y = fitted(mod4), color = "Richards")) +
  labs(x = "", y = "Recovered", color = "Model") +
  scale_color_manual(values = cols) +
  scale_y_continuous(breaks = seq(0, coef(mod2)[1], by = 1000),
                     minor_breaks = seq(0, coef(mod2)[1], by = 500)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

last_plot() +
  scale_y_continuous(trans = "log10", limits = c(10,NA)) +
  labs(y = "Recovered (log10 scale)")

Predictions

Point estimates

df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1),
               fit1 = predict(mod1, newdata = df),
               fit2 = predict(mod2, newdata = df),
               fit3 = predict(mod3, newdata = df),
               fit4 = predict(mod4, newdata = df))
ylim = c(0, max(df[,-(1:3)]))
ggplot(data, aes(x = date, y = y)) + 
  geom_point() + 
  geom_line(data = df, aes(x = date, y = fit1, color = "Exponential")) +
  geom_line(data = df, aes(x = date, y = fit2, color = "Logistic")) +
  geom_line(data = df, aes(x = date, y = fit3, color = "Gompertz")) +
  geom_line(data = df, aes(x = date, y = fit4, color = "Richards")) +
  coord_cartesian(ylim = ylim) +
  labs(x = "", y = "Recovered", color = "Model") +
  scale_y_continuous(breaks = seq(0, max(ylim), by = 1000),
                     minor_breaks = seq(0, max(ylim), by = 1000)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  scale_color_manual(values = cols) +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

Prediction intervals

# compute prediction using Moving Block Bootstrap (MBB) for nls
df = data.frame(x = seq(min(data$x), max(data$x)+14))
df = cbind(df, date = as.Date(df$x, origin = data$date[1]-1))

pred1 = cbind(df, "fit" = predict(mod1, newdata = df))
pred1[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod1, df[df$x > max(data$x),])[,2:3]

pred2 = cbind(df, "fit" = predict(mod2, newdata = df))
pred2[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod2, df[df$x > max(data$x),])[,2:3]

pred3 = cbind(df, "fit" = predict(mod3, newdata = df))
pred3[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod3, df[df$x > max(data$x),])[,2:3]

pred4 = cbind(df, "fit" = predict(mod4, newdata = df))
pred4[df$x > max(data$x), c("lwr", "upr")] = predictMBB.nls(mod4, df[df$x > max(data$x),])[,2:3]

# predictions for next day
pred = rbind(subset(pred1, x == max(data$x)+1, select = 2:5),
             subset(pred2, x == max(data$x)+1, select = 2:5),
             subset(pred3, x == max(data$x)+1, select = 2:5),
             subset(pred4, x == max(data$x)+1, select = 2:5))
print(pred, digits = 3)
##           date   fit   lwr   upr
## 45  2020-04-08 29421 26360 32782
## 451 2020-04-08 24832 23930 25674
## 452 2020-04-08 25678 25223 26140
## 453 2020-04-08 25836 25370 26380

ylim = c(0, max(pred2$upr, pred3$upr, na.rm=TRUE))
ggplot(data, aes(x = date, y = y)) + 
  geom_point() +
  geom_line(data = pred1, aes(x = date, y = fit, color = "Exponential")) +
  geom_line(data = pred2, aes(x = date, y = fit, color = "Logistic")) +
  geom_line(data = pred3, aes(x = date, y = fit, color = "Gompertz")) +
  geom_line(data = pred4, aes(x = date, y = fit, color = "Richards")) +
  geom_ribbon(data = pred1, aes(x = date, ymin = lwr, ymax = upr), 
              inherit.aes = FALSE, fill = cols[1], alpha=0.3) +
  geom_ribbon(data = pred2, aes(x = date, ymin = lwr, ymax = upr), 
              inherit.aes = FALSE, fill = cols[2], alpha=0.3) +
  geom_ribbon(data = pred3, aes(x = date, ymin = lwr, ymax = upr),
              inherit.aes = FALSE, fill = cols[3], alpha=0.3) +
  geom_ribbon(data = pred4, aes(x = date, ymin = lwr, ymax = upr),
              inherit.aes = FALSE, fill = cols[4], alpha=0.3) +
  coord_cartesian(ylim = c(0, max(ylim))) +
  labs(x = "", y = "Recovered", color = "Model") +
  scale_y_continuous(breaks = seq(0, max(ylim), by = 5000),
                     minor_breaks = seq(0, max(ylim), by = 1000)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  scale_color_manual(values = cols) +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

Description of evolution

Positive cases and administered swabs

df = data.frame(date = COVID19$data,
                positives = c(NA, diff(COVID19$totale_casi)),
                swabs = c(NA, diff(COVID19$tamponi)))
df$x = as.numeric(df$date) - min(as.numeric(df$date)) + 1
# df$y = df$positives/df$swabs
df$y = df$positives/c(NA, zoo::rollmean(df$swabs, 2))
df = subset(df, swabs > 50)
# DT::datatable(df[,-4], )
ggplot(df, aes(x = date)) + 
  geom_point(aes(y = swabs, color = "swabs"), pch = 19) +
  geom_line(aes(y = swabs, color = "swabs")) +
  geom_point(aes(y = positives, color = "positives"), pch = 0) +
  geom_line(aes(y = positives, color = "positives")) +
  labs(x = "", y = "Number of cases", color = "") +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  scale_color_manual(values = palette()[c(2,1)]) +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

ggplot(df, aes(x = date, y = y)) + 
  geom_smooth(method = "loess", se = TRUE, col = "black") +
  geom_point(col=palette()[4]) + 
  geom_line(size = 0.5, col=palette()[4]) +
  labs(x = "", y = "% positives among admnistered swabs (two-day rolling mean)") +
  scale_y_continuous(labels = scales::percent_format(),
                     breaks = seq(0, 0.5, by = 0.05)) +
  coord_cartesian(ylim = c(0,max(df$y, na.rm = TRUE))) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

Hospitalized and ICU patients

df = data.frame(date = COVID19$data,
                hospital = c(NA, diff(COVID19$totale_ospedalizzati)),
                icu = c(NA, diff(COVID19$terapia_intensiva)))
df$x = as.numeric(df$date) - min(as.numeric(df$date)) + 1
ggplot(df, aes(x = date, y = hospital)) + 
  geom_smooth(method = "loess", se = TRUE, col = "black") +
  geom_point(col = "orange") + 
  geom_line(size = 0.5, col = "orange") +
  labs(x = "", y = "Change hospitalized patients") +
  coord_cartesian(ylim = range(df$hospital, na.rm = TRUE)) +
  scale_y_continuous(minor_breaks = seq(min(df$hospital, na.rm = TRUE),
                                        max(df$hospital, na.rm = TRUE), 
                                        by = 100)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))

ggplot(df, aes(x = date, y = icu)) + 
  geom_smooth(method = "loess", se = TRUE, col = "black") +
  geom_point(col = "red2") + 
  geom_line(size = 0.5, col = "red2") +
  labs(x = "", y = "Change ICU patients") +
  coord_cartesian(ylim = range(df$icu, na.rm = TRUE)) +
  scale_y_continuous(minor_breaks = seq(min(df$icu, na.rm = TRUE), 
                                        max(df$icu, na.rm = TRUE), 
                                        by = 10)) +
  scale_x_date(date_breaks = "2 day", date_labels =  "%b%d",
               minor_breaks = "1 day") +
  theme_bw() +
  theme(legend.position = "top",
        axis.text.x = element_text(angle=60, hjust=1))